Estimating Welfare in Insurance Markets

Overview

• Objective: Quantify the welfare loss in insurance markets with (adverse or advantageous) selection due to inefficient pricing.
• Key Findings:
  • Evidence of adverse selection in employer-provided health insurance.
  • Quantitative welfare loss of inefficient pricing is small.
• Methodology:
  • Use of price variation to identify and estimate demand and insurers’ costs.

Overview

• Contribution:
  • Show how to use standard consumer and producer theory to empirically estimate welfare losses in markets with selection.
  • According to authors: Fewer assumptions about consumers’ preferences, straightforward implementation, and general

Introduction

• Complementary approach to empirical welfare analysis requiring fewer assumptions
  • Revealed preference, demand and cost curves are sufficient statistics for welfare analysis
• Fewer assumptions at the cost of limiting welfare analysis to only observed (existing) contracts
  • Distortion of offered contracts set not allowed

Introduction

• In Graphical analysis, the authors show that the slope of the cost curve provides a direct test of selection
  • Constant (horizontal) marginal cost curve, no selection; negative, adverse selection; positive, advantageous selection

Introduction

• Employer-provided health insurance
• Individual-level data from Alcoa, Inc., large multinational aluminum producer
  • Health insurance options, choices, and medical insurance claims
  • Price variation used in identification arises due to Alcoa’s organizational structure, (orthogonal to employee characteristics)
• Authors detect adverse selection and estimate the welfare loss to be small (<$10 per employee, 3% of efficient total surplus)

Theoretical Framework

• Individuals choose from two insurance contracts, \(H\) high (insurance) and \(L\) low coverage (no insurance).
• Characteristics of insurance contracts are taken as given, but price is determined endogenously
• Population defined by a distribution \(G(\zeta)\), \(\zeta\) vector of consumer characteristics
• Relative price of contract H by \(p\)

Theoretical Framework

• \(v^H(\zeta_i,p)\) and \(v^L(\zeta_i)\) denote the consumer \(i's\) utility from buying contracts \(H\) and \(L\), respectively
  • \(v^H(\zeta_i,p)\) strictly decreasing in \(p\) and \(v^H(\zeta_i,p=0)>v^H(\zeta_i)\)
• Expected monetary cost of providing insurance to individual \(i\), \(c(\zeta_i)\)

Demand for Insurance

• Each individual makes a discrete choice, buying insurance or not
  • Demand is only a function of price (abstracting from coverage level)
  • Firms cannot offer different prices to different individuals (empirical application: exogenous price variation)
  • Individuals buy at the lowest price available, demand can be sufficiently characterized as a function of the lowest price \(p\)
• Individual \(i\) buys insurance if and only if \(v^H(\zeta_i,p)>v^L(\zeta_i)\)
• Willingness to pay \(\pi(\zeta_i)\equiv\max\{p:v^H(\zeta_i,p)>v^L(\zeta_i) \}\)

Demand for Insurance

• Aggregate demand is \[ D(p)=\int \mathbb{1}(\pi(\zeta)\ge p)dG(\zeta)=\text{Pr}(\pi(\zeta)\ge p) \qquad(1)\]

Supply

• \(N\ge2\) insurance providers set prices à la Bertrand (perfect competition) Under perfect competition, any inefficiency is attributed to selection
• The cost of providing contract \(H\) to individual \(i\) are the insurable cost \(c(\zeta_i)\) No other admin or production costs

Supply

• The average (expected) cost curve \[ \text{AC}(p)=\frac{1}{D(p)}\int c(\zeta)\mathbb{1}(\pi(\zeta)\ge p)dG(\zeta)=\mathbb{E}[c(\zeta)| \pi(\zeta)\ge p] \qquad(2)\]

• The marginal (expected) cost curve is \[ \text{MC}(p)=\mathbb{E}[c(\zeta)|\pi(\zeta)=p] \qquad(3)\]

Equilibrium

• It is profitable to provide insurance
• Single crossing between MC\((p)\) and the demand curve
• The equilibrium \[ p^*=min\{p:p=\text{AC}(p)\} \]

Measuring Welfare

• Authors measure consumer surplus by the certainty equivalent
  • The certainty equivalent of an uncertain outcome is the amount that would make an individual indifferent between obtaining this amount for sure and obtaining the uncertain outcome
• Income effects are ignored (equivalent to assuming CARA utility function)
• \(e^H(\zeta_i)\) and \(e^L(\zeta_i)\) denote the certainty equivalent for consumer \(i\) of an allocation of contract \(H\) and \(L\), respectively
• The willingness to pay \(\pi(\zeta_i)=e^H(\zeta_i)-e^L(\zeta_i)>0\)

Measuring Welfare

• Consumer welfare \[ \text{CS}=\int[(e^H(\zeta)-p)\mathbb{1}(\pi(\zeta)\ge p)+e^L(\zeta)\mathbb{1}(\pi(\zeta)<p)]dG(\zeta) \]
• Producer welfare \[ \text{PS}=\int(p-c(\zeta))\mathbb{1}(\pi(\zeta)\ge p)dG(\zeta) \]

Measuring Welfare

• Total welfare \[ \text{TS=CS+PS}=\int[(e^H(\zeta)-c(\zeta))\mathbb{1}(\pi(\zeta)\ge p)+e^L(\zeta)\mathbb{1}(\pi(\zeta)<p)]dG(\zeta) \]

Graphical Representation

• Adverse selection: Individuals with the highest willingness to pay for insurance are those who, on average, have the highest expected costs
  • Downward-sloping MC: MC is decreasing in quantity
  • As the price falls, the marginal individuals who select contract \(H\) have a lower expected cost than inframarginal individuals, leading to lower average costs
  • Uniform pricing (firms cannot discriminate individuals based on their privately known marginal costs) implies average-cost pricing in equilibrium
  • AC > MC leads to underinsurance in adverse selection

Adverse selection implies the marginal cost curve is downward- sloping, people with the highest willingness to pay also have the highest expected cost to the insurer. Competitive equilibrium is given by point C (demand =average cost), whereas the efficient allocation is E (demand = marginal cost). The (shaded) triangle CDE represents the welfare cost from underinsurance.

Advantageous Selection

• Individuals with the highest willingness to pay insurance have, on average, the lowest expected costs.
  • MC and AC curves are upward-sloping
  • Leading to over-insurance
  • Intuitively, insurance providers have an additional incentive to reduce price, as the inframarginal customers whom they acquire as a result are relatively good risks.

Advantageous selection implies the marginal cost curve is upward- sloping, people with the highest willingness to pay also have the lowest expected cost to the insurer. Competitive equilibrium is given by point C (demand =average cost), whereas the efficient allocation is E (demand = marginal cost). The (shaded) triangle CDE represents the welfare cost from overinsurance.

Sufficient Statistics

• Graphical analyses illustrate that the demand and cost curves are sufficient statistics for welfare analysis of equilibrium and nonequilibrium pricing of existing contracts
  • Different underlying primitives have the same welfare implications if they generate the same demand and cost curves
  • Primitives: preferences and private information summarized by \(\zeta\)

Sufficient Statistics

• This suggests an empirical approach to remain agnostic about the underlying primitives: estimating demand and cost curves
  • Requiring: Revealed preferences choices from individuals can be used for welfare analysis
  • Limits counterfactuals to only price variation for the set of observed insurance contracts
  • Examples: Mandatory insurance coverage, taxes and subsidies for insurance, regulation eliminating an existing contract, etc.

Estimation

• The framework requires data allowing the estimation of the demand curve \(D(p)\) and the average cost curve \(AC(p)\)
• Marginal cost curve can be derived by \[ \begin{aligned} MC(p) &=\frac{\partial TC(p)}{\partial D(p)}=\frac{\partial(AC(p)\cdot D(p))}{\partial D(p)}\\ &=\left(\frac{\partial D(p)}{\partial p}\right)^{-1}\frac{\partial(AC(p)\cdot D(p))}{\partial p} \end{aligned} \]

Identification

• To identify demand, price variation exogenous to unobservable demand characteristics is needed
• Authors claim they do not need an additional source of variation to estimate the average cost curve AC(p) because
  • Expected costs are likely to affect demand, any exogenous price variation to demand is also exogenous to insurable cost

Subsample

• Analysis restricted to a subsample of employees for whom the pricing variation is cleaner
• 3,779 salaried employees who chose one of the two modal health insurance choices: a higher and a lower level of PPO coverage

Price Variation

• In 2004, as part of the new benefit design, company headquarters offered a set of seven different possible pricing menus for employee benefits.
• The coverage options are the same across all the menus, but the prices (employee premiums) associated with these options vary
  • The object of interest is the incremental (annual) premium for contract \(H\) relative to \(L\), \(p=p_H-p_L\)
  • There were six different values of \(p\) in 2004, from $384 to $659

Price Variation by Business Unit

• The president of each business unit determined the price menu for the employees of their respective business unit
• Alcoa had forty business units. Each unit has essentially complete independence to run its business
• Business units are typically organized by functionality and are independent of geography (there are often multiple business units in the same state)

Price Variation by Business Unit

• The median business unit has about 500 employees
• The business unit president may choose different price menus for employees within his unit based on their location and employment type (salaried/hourly, hourly by union)
• As a result, employees doing the same job in the same location may face different prices for their health insurance benefits due to their business affiliations

Price Variation

Price Variation Analysis

• Authors claim that they could not reject the null that all the coefficients are jointly uncorrelated; F-test p-value=0.14
• Business unit presidents were opposed to charging employees much for health insurance coverage
• After 2004, Alcoa no longer gave business unit presidents a choice on benefit prices and adopted a uniform pricing structure
• A similar analysis for hourly workers reveals statistically significant differences across employees who face different prices

Price Variation Analysis

Empirical Strategy

• \(p_i=p_i^H-p_i^L\), relative price employee \(i\) faces
• \(D_i\) is equal to 1 if employee \(i\) chooses contract \(H\), and O if \(L\)
• \(m_i\) total medical expenditure of employee \(i\) and covered family members
• \(c_i=c(m_i;H)-c(m_i;L)\) is the incremental cost to the insurer from providing contract H relative to L, holding \(m_i\) fixed.
  • \(c(m_i;H)\) and \(c(m_i;L)\) denote the cost to the insurance company from medical expenditures under H and L
  • \(c(m_i;H)\) is observed; \(c(m_I;L)\) is computed counterfactually using the claims data and the plan L rules

Estimating Equations

\[ \begin{aligned} D_i&=\alpha+\beta p_i + \epsilon_i \\ c_i&=\gamma+\delta p_i + u_i \end{aligned} \qquad(4)\]

MC and Equilibrium P and Q

Having estimated the demand and AC, we can find MC,

\[ MC(p)=\frac{\alpha\delta}{\beta}+\gamma+2\delta p \]

With MC, we can find the equilibrium and the efficient price and quantity; Equilibrium AC(p)=p, Efficient MC(p)=p \[ \begin{aligned} P_{eq} &=\gamma/(1-\delta)\\ Q_{eq} &=\alpha + \beta(\gamma/(1-\delta))\\ \end{aligned} \]

Efficient P and Q

Efficient MC(p)=p \[ \begin{aligned} P_{eff} &=\frac{1}{(1-2\delta)(\frac{\alpha\delta}{\beta}+\gamma)}\\ Q_{eff} &=\alpha+\frac{1}{(1-2\delta)(\alpha\delta+\beta\gamma)} \end{aligned} \]

Welfare Loss \[ \Delta_{CDE}=-\frac{\delta^2}{2(1-2\delta)\beta}\left(\alpha+\frac{\beta\gamma}{1-\delta}\right)^2 \]

Results

Results

Results

• Downward sloping demand \(\beta=-0.0007\)
  • a $100 increase in price reduces the probablity of getting \(H\) by 11%
• Cost curve slope indicates the existence of adverse selection \(\delta=0.155\)
  • The average cost of individuals who purchased contract \(H\) is higher when the price is higher
  • The average cost curve is therefore downward-sloping in quantity

Results

• Welfare cost of adverse selection $9.55 per employee per year
• Adverse selection increases price by almost $200 and reduces the share of H by 14 p.p.